operator theory: advances and applications vol. 207 founded in 1979 by israel gohberg editors:...

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Operator Theory: Advances and Applications

Vol. 207

Founded in 1979 by Israel Gohberg

Editors:Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel)Marinus A. Kaashoek (Amsterdam, The Netherlands)Heinz Langer (Vienna, Austria)Christiane Tretter (Bern, Switzerland)

Associate Editors:Vadim Adamyan (Odessa, Ukraine)Albrecht Böttcher (Chemnitz, Germany)B. Malcolm Brown (Cardiff, UK)Raul Curto (Iowa City, IA, USA)Fritz Gesztesy (Columbia, MO, USA)Pavel Kurasov (Lund, Sweden)Leonid E. Lerer (Haifa, Israel)Vern Paulsen (Houston, TX, USA)Mihai Putinar (Santa Barbara, CA, USA)Leiba Rodman (Williamsburg, VI, USA)Ilya M. Spitkovsky (Williamsburg, VI, USA)

Honorary and Advisory Editorial Board:Lewis A. Coburn (Buffalo, NY, USA)Ciprian Foias (College Station, TX, USA)J. William Helton (San Diego, CA, USA)Thomas Kailath (Stanford, CA, USA)Peter Lancaster (Calgary, AB, Canada)Peter D. Lax (New York, NY, USA)Donald Sarason (Berkeley, CA, USA)Bernd Silbermann (Chemnitz, Germany)Harold Widom (Santa Cruz, CA, USA)

SubseriesLinear Operators and Linear SystemsSubseries editors:Daniel Alpay (Beer Sheva, Israel)Birgit Jacob (Wuppertal, Germany)André C.M. Ran (Amsterdam, The Netherlands)

SubseriesAdvances in Partial Differential EquationsSubseries editors:Bert-Wolfgang Schulze (Potsdam, Germany)Michael Demuth (Clausthal, Germany)Jerome A. Goldstein (Memphis, TN, USA)Nobuyuki Tose (Yokohama, Japan)Ingo Witt (Göttingen, Germany)


A Glimpse at Hilbert Space OperatorsPaul R. Halmos in Memoriam

Sheldon AxlerPeter RosenthalDonald Sarason Editors

Birkhäuser


2010 Mathematics Subject Classification: Primary 47A, 47B; secondary 46

Library of Congress Control Number: 2010927369

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-0346-0346-1

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

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ISBN 978-3-0346-0346-1 e-ISBN 978-3-0346-0347-8

9 8 7 6 5 4 3 2 1 www.birkhauser.ch

Editors:

Sheldon AxlerDepartment of Mathematics San Francisco State UniversitySan Francisco, CA 94132-3163USAe-mail: [email protected]

Peter RosenthalDepartment of Mathematics University of TorontoToronto, ON MS5 2E4Canadae-mail: [email protected]

Donald SarasonDepartment of MathematicsUniversity of California Berkeley, CA 94720-3840USAe-mail: [email protected]


Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Part I: Paul Halmos

V.S. SunderPaul Halmos – Expositor Par Excellence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

J. EwingPaul Halmos: In His Own Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

H. Radjavi and P. RosenthalObituary: Paul Halmos, 1916–2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

G. PiranianMathematical Review of “How to Write Mathematics”by P.R. Halmos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Publications of Paul R. Halmos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Photos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Part II: Articles

J. Agler and J.E. McCarthyWhat Can Hilbert Spaces Tell Us About Bounded Functionsin the Bidisk? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

W. ArvesonDilation Theory Yesterday and Today . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

S. AxlerToeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

H. BercoviciDual Algebras and Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

J.B. Conway and N.S. FeldmanThe State of Subnormal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

R. Curto and M. PutinarPolynomially Hyponormal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195


vi Contents

K.R. DavidsonEssentially Normal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

M.A. Dritschel and J. RovnyakThe Operator Fejér-Riesz Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

P.S. MuhlyA Halmos Doctrine and Shifts on Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 255

V.V. PellerThe Behavior of Functions of Operators Under Perturbations . . . . . . . . 287

G. PisierThe Halmos Similarity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

H. Radjavi and P. RosenthalPaul Halmos and Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

D. SarasonCommutant Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

V.S. SunderDouble Cones are Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359


Preface

Paul Richard Halmos, who lived a life of unbounded devotion to mathematics andto the mathematical community, died at the age of 90 on October 2, 2006. Thisvolume is a memorial to Paul by operator theorists he inspired.

Paul’s initial research, beginning with his 1938 Ph.D. thesis at the Universityof Illinois under Joseph Doob, was in probability, ergodic theory, and measuretheory. A shift occurred in the 1950s when Paul’s interest in foundations led himto invent a subject he termed algebraic logic, resulting in a succession of paperson that subject appearing between 1954 and 1961, and the book Algebraic Logic,published in 1962.

Paul’s first two papers in pure operator theory appeared in 1950. After 1960Paul’s research focused on Hilbert space operators, a subject he viewed as encom-passing finite-dimensional linear algebra.

Beyond his research, Paul contributed to mathematics and to its communityin manifold ways: as a renowned expositor, as an innovative teacher, as a tirelesseditor, and through unstinting service to the American Mathematical Society andto the Mathematical Association of America. Much of Paul’s influence flowed at apersonal level. Paul had a genuine, uncalculating interest in people; he developedan enormous number of friendships over the years, both with mathematicians andwith nonmathematicians. Many of his mathematical friends, including the editorsof this volume, while absorbing abundant quantities of mathematics at Paul’s knee,learned from his advice and his example what it means to be a mathematician.

The first section of this volume contains three tributes to Paul written on theoccasion of his death. They elaborate on the brief remarks in the preceding para-graph, and are reproduced here with the kind permission of their authors and theiroriginal publishers. The last item in the first section reproduces the late GeorgePiranian’s Mathematical Review of Paul’s article How to write mathematics. A listof Paul’s publications comprises the next section, which is followed by a section ofphotographs of Paul and photographs taken by Paul.

The main and final section consists of a collection of expository articles byprominent operator theorists. From these articles, this generation of operator the-orists and future generations will get a glimpse of many aspects of their subject,and of how Paul enriched and advanced it through his fundamental insights andprescient questions.


viii Preface

Acknowledgement

We thank the authors of the expository articles for the high quality of their expo-sitions.

We thank Thomas Hempfling, Birkhäuser’s Executive Editor for Mathemat-ics, for his constant support and sound advice.

We thank Mary Jennings for volunteering her expertise in assembling thephoto section. This was an arduous task that Mary carried out with painstakingcare, offering helpful suggestions along the way. (Other photo acknowledgementsare in the introduction to the photo section.)

This volume was first proposed by the late Israel Gohberg, the founder anduntil his recent death the chief editor of the series in which it appears. With Israel’spassing, operator theory has lost another of its giants.

Sheldon Axler, Peter Rosenthal, Donald Sarason

December 2009


Part I

Paul Halmos


Paul Halmos, circa 1980


Operator Theory:Advances and Applications, Vol. 207, 3–10c© 2010 Springer Basel AG

Paul Halmos – Expositor Par Excellence*

V.S. Sunder

Abstract. Paul Richard Halmos, one of the best expositors of mathematics– be it with pen on paper or with chalk on blackboard – passed away onOctober 2, 2006 after a brief period of illness. This article is an attempt topay homage to him by recalling some of his contributions to mathematics.

Mathematics Subject Classification (2000). 47-xx.

Keywords. Halmos, Hilbert space, operator theory, ergodic theory.

Introduction

Here is what Donald Sarason – arguably the most accomplished PhD student ofHalmos – writes about his extraordinary teacher (in [1]):

“Halmos is renowned as an expositor. His writing is some thing he works hardat, thinks intensely about, and is fiercely proud of. (Witness: “How to write math-ematics” (see [2]).) In his papers, he is not content merely to present proofs thatare well organized and clearly expressed; he also suggests the thought processesthat went into the construction of his proofs, pointing out the pitfalls he encoun-tered and indicating helpful analogies. His writings clearly reveal his commitmentas an educator. In fact, Halmos is instinctively a teacher, a quality discernible inall his mathematical activities, even the most casual ones.

Most of us, when we discover a new mathematical fact, how ever minor, areusually eager to tell someone about it, to display our cleverness. Halmos behavesdifferently: he will not tell you his discovery, he will ask you about it, and challengeyou to find a proof. If you find a better proof than his, he will be delighted, becausethen you and he will have taught each other.

To me, Halmos embodies the ideal mixture of researcher and teacher. Inhim, each role is indistinguishable from the other. Perhaps that is the key to hisremarkable influence.”

* Reprinted with permission from the February 2007 issue of Resonance.


4 V.S. Sunder

Many of his expository writings, elaborating on his views on diverse topics– writing, lecturing, and doing mathematics – are a ‘must read’ for every seriousstudent of mathematics. Conveniently, many of them have been collected togetherin [2].

And here is what one finds in the web pages of the Mathematics Associationof America (MAA):

“Professor Halmos was a famed author, editor, teacher, and speaker of dis-tinction. Nearly all of his many books are still in print. His Finite DimensionalVector Spaces, Naive Set Theory, Measure Theory, Problems for MathematiciansYoung and Old, and I Want to be a Mathematician are classic books that reflect hisclarity, conciseness, and color. He edited the American Mathematical Monthly from1981–1985, and served for many years as one of the editors of the Springer-Verlagseries Undergraduate Texts in Mathematics and Graduate Texts in Mathematics.”

While Halmos will be the first to acknowledge that there were far more ac-complished mathematicians around him, he would at the same time be the lastto be apologetic about what he did. There was the famous story of how, as ayoung and very junior faculty member at the University of Chicago, he would notlet himself be bullied by the very senior faculty member André Weil on a matterof faculty recruitment. His attitude – which functional analysts every where cando well to remember and take strength from – was that while algebraic geometrymight be very important, the usefulness of operator theory should not be denied.

Even in his own area of specialisation, there were many mathematicians morepowerful than he; but he ‘had a nose’ for what to ask and which notions to concen-trate on. The rest of this article is devoted to trying to justify the assertion of thelast line and describing some of the mathematics that Halmos was instrumentalin creating. Also, the author has attempted to conform with Halmos’ tenet thatsymbols should, whenever possible, be substituted by words, in order to assist thereader’s assimilation of the material. An attempt to write a mathematical articlesubject to this constraint will convince the reader of the effort Halmos put intohis writing!

The invariant subspace problem

Although Halmos has done some work in probability theory (his PhD thesis waswritten under the guidance of the celebrated probabilist J.L. Doob), statistics(along with L.J. Savage, he proved an important result on sufficient statistics),ergodic theory and algebraic logic, his preferred area of research (where he eventu-ally ‘settled down’) was undoubtedly operator theory, more specifically, the studyof bounded operators on Hilbert space. Recall that a Hilbert space means a vectorspace over the field of complex numbers which is equipped with an inner productand is complete with respect to the norm arising from the inner product. Most ofhis research work revolved around the so-called invariant subspace problem, whichasks: Does every bounded (= continuous) linear operator on a Hilbert space (of


Paul Halmos – Expositor Par Excellence 5

dimension at least 2) admit a non-trivial invariant subspace, meaning: Is therea closed subspace, other than the zero subspace and the whole space (the twoextreme trivial ones1) which is mapped into itself by the operator? The answeris negative over the field of real numbers (any rotation in the plane yielding acounterexample), and is positive in the finite-dimensional complex case (thanks tocomplex matrices having complex eigenvalues).

The first progress towards the solution of this problem came when von Neu-mann showed that if an operator is compact (i.e., if it maps the unit ball into acompact set, or equivalently, if it is uniformly approximable on the unit ball byoperators with finite-dimensional range), then it does indeed have a non-trivialinvariant subspace. This was later shown, by Aronszajn and Smith, to continue tobe true for compact operators over more general Banach, rather than just Hilbert,spaces.

Then Smith asked and Halmos publicized the question of whether an operatorwhose square is compact had invariant subspaces. It was subsequently shown byBernstein and Robinson, using methods of ‘non-standard analysis’, that if somenon-zero polynomial in an operator is compact, then it has invariant subspaces.Very shortly after, Halmos came up with an alternative proof of this result, usingstandard methods of operator theory.

Quasitriangularity, quasidiagonality and theWeyl-von Neumann-Voiculescu theorem

Attempting to isolate the key idea in the proof of the Aronszajn-Smith theorem,Halmos identified the notion of quasitriangular operators. ‘Triangular’ operators –those which possess an upper triangular matrix with respect to some orthonormalbasis – may also be described (since finite-dimensional operators are triangular) asthose which admit an increasing sequence of finite-dimensional invariant subspaceswhose union is dense in the Hilbert space. Halmos’ definition of quasitriangularityamounts to weakening ‘invariant’ to ‘asymptotically invariant’ in the previous sen-tence. An entirely equivalent requirement, as it turns out, is that the operator isof the form ‘triangular + compact’; and the search was on for invariant subspacesof quasitriangular operators. This was until a beautiful ‘index-theoretic’ charac-terisation of quasitriangularity was obtained by Apostol, Foias and Voiculescu,which had the unexpected consequence that if an operator or its adjoint is notquasitriangular, then it has a non-trivial invariant subspace.

There is a parallel story involving quasidiagonality, also starting with a def-inition by Halmos and ending with a spectacular theorem of Voiculescu. Recallthat in finite-dimensional Hilbert spaces, according to the spectral theorem, self-adjoint operators have diagonal matrices with respect to some orthonormal ba-sis, and two selfadjoint operators are unitarily equivalent precisely when they

1This is why the Hilbert space needs to be at least 2-dimensional.


6 V.S. Sunder

have the same eigenvalues (i.e., diagonal entries in a diagonal form) which oc-cur with the same multiplicity. Thus the spectrum of an operator (i.e., the set(T ) = {λ ∈ C : (T − λ) is not invertible}) and the associated spectral multiplicity(the multiplicity of λ is the dimension of the nullspace of T − λ) form a completeset of invariants for unitary equivalence in the class of self-adjoint operators.

In the infinite-dimensional case, Hermann Weyl proved that the so-called‘essential spectrum’ of a self-adjoint operator is left unchanged when it is perturbedby a compact operator; (here, the ‘essential spectrum’ of a self-adjoint operator isthe complement, in the spectrum, of ‘isolated eigenvalues of finite multiplicity’;)while von Neumann showed that the essential spectrum is a complete invariantfor ‘unitary equivalence modulo compact perturbation’ in the class of self-adjointoperators. Thus, if one allows compact perturbations, spectral multiplicity is nolonger relevant. It follows that self-adjoint operators are expressible in the form‘diagonal + compact’; von Neumann even proved the strengthening with ‘compact’replaced by ‘Hilbert-Schmidt’. (Recall that an operator T is said to be a Hilbert-Schmidt operator if

∑ ‖Ten‖2 ≤ ∞ for some (equivalently every) orthonormalbasis {en} of the Hilbert space.)

Halmos asked if both statements had valid counterparts for normal opera-tors; specifically, does every normal operator admit a decomposition of the form(a) diagonal + Hilbert-Schmidt, and less stringently (b) diagonal + compact. Bothquestions were shown to have positive answers as a consequence of the brilliant‘noncommutative Weyl von Neumann theorem’ due to Voiculescu (about repre-sentations of C∗-algebras, which specialises in the case of commutative ∗-algebrasto the desired statements about normal operators); however(a) had also been in-dependently settled by I.D. Berg.

Subnormal operators and unitary dilations

There were two other major contributions to operator theory by Halmos: subnor-mal operators and unitary dilations. Both were born of his unwavering belief thatthe secret about general operators lay in their relationship to normal operators.He defined a subnormal operator to be the restriction of a normal operator toan invariant subspace; the most striking example is the unilateral shift. (Recallthat the bilateral shift is the clearly unitary, hence normal, operator on the bi-lateral sequence space �2(Z) = {f : Z → C : ∑n∈Z |f(n)|2 < ∞} defined by theequation (Wf)(n) = f(n − 1). In the previous sentence, if we replace Z by Z+,the analogous equation defines the unilateral shift U on the one-sided sequencespace �2(Z+), which is the prototypical isometric operator which is not unitary. Itshould be clear that �2(Z+) may be naturally identified with a subspace of �2(Z)which is invariant under W , and that the restriction of W to that subspace maybe identified with U .) Halmos proved that a general subnormal operator exhibitsmany properties enjoyed by this first example. For instance, he showed that thenormal extension of a subnormal operator is unique under a mild (and natural)



minimality condition. (The minimal normal extension of the unilateral shift is thebilateral shift.) Halmos also established that the spectrum of a subnormal oper-ator is obtained by ‘filling in some holes’ in the spectrum of its minimal normalextension. Many years later, Scott Brown fulfilled Halmos’ hope by establishingthe existence of non-trivial invariant subspaces of a subnormal operator.

More generally than extensions, Halmos also initiated the study of dilations.It is best to first digress briefly into operator matrices. The point is that if T isan operator on H, then any direct sum decomposition H = H1 ⊕ H2 leads to anidentification

T =[T11 T12T21 T22

],

where Tij : Hj → Hi; 1 ≤ i, j ≤ 2 are operators which are uniquely determined bythe requirement that if the canonical decomposition H � x = x1 + x2, xi ∈ Hi iswritten as x = [ x1x2 ], then

T

[x1x2

]=[T11 T12T21 T22

] [x1x2

]=[T11x1 + T12x2T21x1 + T22x2

].

Thus, for instance, the orthogonal projection P1 of H onto H1 is given by

P1 =[idH1 00 0

]and T11 = P1T |H1 . It is customary to call T11 the compression of T to H1 and tocall T a dilation of T11. (Note that if, and only if, T21 = 0, then ‘compression’ and‘dilation’ are nothing but ‘restriction’ and ‘extension’.) Halmos wondered, but notfor long, if every operator had a normal dilation; he proved that an operator hasa unitary dilation if (and only if) it is a contraction (i.e., maps the unit ball of theHilbert space into itself).

Subsequently, Sz.-Nagy showed that every contraction in fact has a ‘powerdilation’: i.e., if T is a contraction, then there is a unitary operator U on someHilbert space such that, simultaneously, Un is a dilation of Tn for every n ≥ 0.Halmos noticed that this established the equivalence of the following conditions:• T is a contraction,• T n is a contraction, for each n,• ‖p(T )‖ ≤ sup{|f(z)| : z ∈ D (i.e., |z| < 1)},

and asked if the following conditions were equivalent:• T is similar to a contraction, i.e., there exists an invertible operator S such

that S−1TS is a contraction.• supn ‖T n‖ ≤ K.• ‖p(T )‖ ≤ K sup{|f(z)| : z ∈ D}.

This question, as well as generalisations with D replaced by more general domainsin Cn, had to wait a few decades before they were solved by Gilles Pisier us-ing ‘completely bounded maps’ and ‘operator spaces’ which did not even exist inHalmos’ time!


8 V.S. Sunder

His influence on operator theory may be gauged by the activity in this areaduring the period between his two expository papers (which are listed as his ‘Tech-nical papers’ at the end of this article, and which may both be found in [1]).

He listed 10 open problems in the area in the first paper, and reviewed theprogress made, in the second paper. While concluding the latter paper, he wrote:

I hope that despite its sins of omission, this survey conveyed the flavorand the extent of progress in the subject during the last decade.

Likewise, I hope I have been able to convey something of the brilliance ofthe expositor in Halmos and the excitement and direction he brought to operatortheory in the latter half of the last century.

Ergodic theory

Although the above account mainly discusses Halmos’ contributions to operatortheory, undoubtedly due to limitations of the author’s familiarity with the areasin which Halmos worked, it would be remiss on the author’s part to not make atleast passing mention of his contributions to ergodic theory.

He wrote the first English book on ergodic theory (the first book on thesubject being Hopf’s, in German). He made his influence felt in the field throughthe problems he popularised and the investigations he undertook. For instance,he gave a lot of publicity, through his book, to the question of whether a ‘non-singular’ transformation of a measure space – i.e., one which preserved the class ofsets of measure zero – admitted an equivalent σ-finite measure which it preserved.This led to the negative answer by Ornstein, subsequent results in the area by Ito,Arnold, and others and culminated in the very satisfying results by Krieger on orbitequivalence. Other contributions of his include the consideration of topologies onthe set of measure-preserving transformations of a measure space (influenced nodoubt by the ‘category-theoretic’ results obtained by Oxtoby and Ulam about theampleness of ergodic homeomorphisms among all homeomorphisms of a cube in ndimensions) and initiating the search for square roots (and cube roots, etc.) of anergodic transformation.

A brief biography

A brief non-mathematical account of his life follows. For a more complete andeminently readable write-up which serves the same purpose (and much more at-tractively, with numerous quotes of Halmos which serve to almost bring him tolife), the reader is advised to look at the web-site:

http://scidiv.bcc.ctc.edu/Math/Halmos.html

Halmos’ life was far from ‘routine’ – starting in Hungary and quickly movingto America. The following paragraph from his autobiographical book [8] containsa very pithy summing up of his pre-America life:



My father, a widower, emigrated to America when I, his youngest son,was 8 years old. When he got established, he remarried, presented uswith two step-sisters, and began to import us: first my two brothers, andlater, almost immediately after he became a naturalized citizen, myself.In view of my father’s citizenship I became an instant American themoment I arrived, at the age of 13.

The automathography referred to in [8] contains other vignettes where wecan see the problems/difficulties the young Halmos faced in coping with an alienlanguage and culture and a periodically unfriendly ‘goddam foreigner’ attitude.

After a not particularly spectacular period of undergraduate study, he beganby studying philosophy and mathematics, hoping to major in the former. Fortu-nately for the thousands who learnt linear algebra, measure theory and Hilbertspace theory through his incomparable books, he fared poorly in the oral compre-hensive exam for the masters’ degree, and switched to mathematics as a major.It was only later, when he interacted with J.L. Doob that he seems to have be-come aware of the excitement and attraction of mathematics; and wrote a thesison Invariants of Certain Stochastic Transformations: The Mathematical Theoryof Gambling Systems.

After he finished his PhD in 1938, he “typed 120 letters of application, andgot two answers: both NO.” “The U of I took pity on me and kept me on as aninstructor.” In the middle of that year a fellow graduate student and friend (WarrenAmbrose) of Halmos received a fellowship at the Institute for Advanced Study,Princeton. “That made me mad. I wanted to go, too! So I resigned my teachingjob, borrowed $ 1000 from my father, [wrangled] an unsupported membership (= aseat in the library) at the Institute, and moved to Princeton.”

There, he attended courses, including the one by John von Neumann (Every-body called him Johnny) on ‘Rings of Operators’. Von Neumann’s official assistantwho was more interested in Topology, showed von Neumann the notes that Hal-mos was taking of the course, and Halmos became the official note-taker for thecourse and subsequently became von Neumann’s official assistant. The next year,“with no official pre-arrangement, I simply tacked up a card on the bulletin boardin Fine Hall saying that I would offer a course called ‘Elementary theory of ma-trices’, and I proceeded to offer it.” About a dozen students attended the course,some took notes and these notes were subsequently pruned into what becameFinite-dimensional vector spaces; and Halmos’ career and book-writing skills wereoff and running.

As a personal aside, this book was this author’s first introduction to thecharm of abstract mathematics, and which prompted him to go to graduate schoolat Indiana University to become Halmos’ PhD student – his last one as it turnedout. This author cannot begin to enumerate all the things he learnt from thissupreme teacher, and will forever be in his debt.

As a final personal note, I should mention Virginia, his warm and hospitablewife since 1945. I remember going to their house for lunch once and finding Paul


10 V.S. Sunder

all alone at home; his grumbled explanation: “Ginger has gone cycling to the oldfolks home, to read to some people there, who are about 5 years younger thanher!”. (She was past 70 then.) She still lives at Los Gatos, California. They neverhad children, but there were always a couple of cats in their house.

It seems appropriate to end with this quote from the man himself: “I’m nota religious man, but it’s almost like being in touch with God when you’re thinkingabout mathematics.”

Suggested reading

[1] P.R. Halmos, Selecta – Research Mathematics, Springer-Verlag, New York,1983.

[2] P.R. Halmos, Selecta – Expository Writing, Springer-Verlag, New York, 1983.

Technical papers by Halmos

[3] Ten Problems in Hilbert space, Bulletin of the AMS, Vol. 76, No. 5, pp.887–993, 1970.

[4] Ten Years in Hilbert space, Integral Eqs. and Operator Theory, Vol. 2/4,pp. 529–564, 1979.

Expository articles by Halmos

[5] How to write Mathematics, l’Enseignement mathématique, Vol. XVI, No. 2,pp. 123–152, 1970.

[6] How to write Mathematics, Notices of the AMS, Vol. 23, No. 4, pp. 155–158,1974.

[7] The Teaching of Problem Solving, Amer. Math. Monthly, Vol. 82, No. 5, pp.466–470, 1975.

Two non-technical books by Halmos

[8] I want to be a Mathematician – an Automathography, Springer-Verlag, NewYork, 1985.

[9] I have a Photographic Memory, AMS, Providence, RI, 1987.

V.S. SunderThe Institute of Mathematical SciencesChennai, 600 113, Indiae-mail: [email protected]



Paul Halmos: In His Own Words*

John Ewing

Abstract. Paul Halmos died on October 2, 2006, at the age of 90. After hisdeath, many people wrote about his career and praised both his mathematicaland his expository skills. Paul would have complained about that: He oftensaid he could smell great mathematicians, and he himself was not one of them.

But he was wrong. He was a master of mathematics in multiple ways,and he influenced mathematicians and mathematical culture throughout hiscareer. Unlike most other master mathematicians, Paul’s legacy was not mere-ly mathematics but rather advice and opinion about mathematical life—writing, publishing, speaking, research, or even thinking about mathematics.Paul wrote about each of these topics with an extraordinary mixture of convic-tion and humility. Mathematicians paid attention to what he wrote, and theyoften quoted it (and still do—“every talk ought to have one proof”). They dis-agreed and frequently wrote rebuttals. They passed along his wisdom to theirstudents, who passed it along to theirs. Paul Halmos’s writing affected the pro-fessional lives of nearly every mathematician in the latter half of the twentiethcentury, and it will continue to influence the profession for years to come.

How does one write about great writing? Explanations of great expo-sition always fall flat, like analyses of great poems or elucidations of famouspaintings. Art is best exhibited, not explained.

And so here is a collection of excerpts from the writing of Paul Halmos,giving advice, offering opinions, or merely contemplating life as a mathe-matician—all in his own words. – J.E.

Mathematics Subject Classification (2000). 00A05, 00B10.

Keywords. Mathematical exposition.

On writing

Excerpts from: “How to write mathematics”, Enseign. Math. (2) 16 (1970),123–152.

. . . I think I can tell someone how to write, but I can’t think who wouldwant to listen. The ability to communicate effectively, the power to be intelligible,

* This article first appeared in Notices of the AMS, 54 (2007), 1136–1144. Reprinted withpermission.


12 J. Ewing

is congenital, I believe, or, in any event, it is so early acquired that by the timesomeone reads my wisdom on the subject he is likely to be invariant under it. Tounderstand a syllogism is not something you can learn; you are either born withthe ability or you are not. In the same way, effective exposition is not a teachableart; some can do it and some cannot. There is no usable recipe for good writing.

Then why go on? A small reason is the hope that what I said isn’t quiteright; and, anyway, I’d like a chance to try to do what perhaps cannot be done.A more practical reason is that in the other arts that require innate talent, eventhe gifted ones who are born with it are not usually born with full knowledge ofall the tricks of the trade. A few essays such as this may serve to “remind” (in thesense of Plato) the ones who want to be and are destined to be the expositors ofthe future of the techniques found useful by the expositors of the past.

The basic problem in writing mathematics is the same as in writing biology,writing a novel, or writing directions for assembling a harpsichord: the problem isto communicate an idea. To do so, and to do it clearly, you must have somethingto say, and you must have someone to say it to, you must organize what you wantto say, and you must arrange it in the order you want it said in, you must writeit, rewrite it, and re-rewrite it several times, and you must be willing to thinkhard about and work hard on mechanical details such as diction, notation, andpunctuation. That’s all there is to it. . .

It might seem unnecessary to insist that in order to say something well youmust have something to say, but it’s no joke. Much bad writing, mathematicaland otherwise, is caused by a violation of that first principle. Just as there are twoways for a sequence not to have a limit (no cluster points or too many), there aretwo ways for a piece of writing not to have a subject (no ideas or too many).

The first disease is the harder one to catch. It is hard to write many wordsabout nothing, especially in mathematics, but it can be done, and the result isbound to be hard to read. There is a classic crank book by Carl Theodore Heisel[The Circle Squared Beyond Refutation, Heisel, Cleveland, 1934] that serves asan example. It is full of correctly spelled words strung together in grammaticalsentences, but after three decades of looking at it every now and then I still cannotread two consecutive pages and make a one-paragraph abstract of what they say;the reason is, I think, that they don’t say anything.

The second disease is very common: there are many books that violate theprinciple of having something to say by trying to say too many things. . .

The second principle of good writing is to write for someone. When youdecide to write something, ask yourself who it is that you want to reach. Are youwriting a diary note to be read by yourself only, a letter to a friend, a researchannouncement for specialists, or a textbook for undergraduates? The problems aremuch the same in any case; what varies is the amount of motivation you need toput in, the extent of informality you may allow yourself, the fussiness of the detailthat is necessary, and the number of times things have to be repeated. All writingis influenced by the audience, but, given the audience, the author’s problem is tocommunicate with it as best he can. . .


Paul Halmos: In His Own Words 13

Everything I’ve said so far has to do with writing in the large, global sense;it is time to turn to the local aspects of the subject.

The English language can be a beautiful and powerful instrument for inter-esting, clear, and completely precise information, and I have faith that the same istrue for French or Japanese or Russian. It is just as important for an expositor tofamiliarize himself with that instrument as for a surgeon to know his tools. Euclidcan be explained in bad grammar and bad diction, and a vermiform appendix canbe removed with a rusty pocket knife, but the victim, even if he is unconscious ofthe reason for his discomfort, would surely prefer better treatment than that. . .

My advice about the use of words can be summed up as follows. (1) Avoidtechnical terms, and especially the creation of new ones, whenever possible. (2)Think hard about the new ones that you must create; consult Roget; and makethem as appropriate as possible. (3) Use the old ones correctly and consistently,but with a minimum of obtrusive pedantry. . .

Everything said about words, applies, mutatis mutandis, to the even smallerunits of mathematical writing, the mathematical symbols. The best notation is nonotation; whenever possible to avoid the use of a complicated alphabetic apparatus,avoid it. A good attitude to the preparation of written mathematical expositionis to pretend that it is spoken. Pretend that you are explaining the subject to afriend on a long walk in the woods, with no paper available; fall back on symbolismonly when it is really necessary.

On speaking

Excerpts from: “How to talk mathematics”, Notices of AMS 21 (1974), 155–158.

What is the purpose of a public lecture? Answer: to attract and to inform.We like what we do, and we should like for others to like it too; and we believe thatthe subject’s intrinsic qualities are good enough so that anyone who knows whatthey are cannot help being attracted to them. Hence, better answer: the purposeof a public lecture is to inform, but to do so in a manner that makes it possiblefor the audience to absorb the information. An attractive presentation with nocontent is worthless, to be sure, but a lump of indigestible information is worthno more. . .

Less is more, said the great architect Mies van der Rohe, and if all lecturersremember that adage, all audiences would be both wiser and happier.

Have you ever disliked a lecture because it was too elementary? I am surethat there are people who would answer yes to that question, but not many. Everytime I have asked the question, the person who answered said no, and then lookeda little surprised at hearing the answer. A public lecture should be simple andelementary; it should not be complicated and technical. If you believe and can acton this injunction (“be simple”), you can stop reading here; the rest of what I haveto say is, in comparison, just a matter of minor detail.


14 J. Ewing

To begin a public lecture to 500 people with “Consider a sheaf of germs ofholomorphic functions. . . ” (I have heard it happen) loses people and antagonizesthem. If you mention the Künneth formula, it does no harm to say that, at least asfar as Betti numbers go, it is just what happens when you multiply polynomials.If you mention functors, say that a typical example is the formation of the dualsof vector spaces and the adjoints of linear transformations.

Be simple by being concrete. Listeners are prepared to accept unstated (buthinted) generalizations much more than they are able, on the spur of the mo-ment, to decode a precisely stated abstraction and to re-invent the special casesthat motivated it in the first place. Caution: being concrete should not lead toconcentrating on the trees and missing the woods. In many parts of mathematicsa generalization is simpler and more incisive than its special parent. (Examples:Artin’s solution of Hilbert’s 17th problem about definite forms via formally realfields; Gelfand’s proof of Wiener’s theorem about absolutely convergent Fourierseries via Banach algebras.) In such cases there is always a concrete special casethat is simpler than the seminal one and that illustrates the generalization withless fuss; the lecturer who knows his subject will explain the complicated specialcase, and the generalization, by discussing the simple cousin.

Some lecturers defend complications and technicalities by saying that that’swhat their subject is like, and there is nothing they can do about it. I am skeptical,and I am willing to go so far as to say that such statements indicate incompleteunderstanding of the subject and of its place in mathematics. Every subject, andeven every small part of a subject, if it is identifiable, if it is big enough to give anhour talk on, has its simple aspects, and they, the simple aspects, the roots of thesubject, the connections with more widely known and older parts of mathematics,are what a non-specialized audience needs to be told.

Many lecturers, especially those near the foot of the academic ladder, anxiousto climb rapidly, feel under pressure to say something brand new—to impress theirelders with their brilliance and profundity. Two comments: (1) the best way to dothat is to make the talk simple, and (2) it doesn’t really have to be done. It maybe entirely appropriate to make the lecturer’s recent research the focal point ofthe lecture, but it may also be entirely appropriate not to do so. An audience’sevaluation of the merits of a talk is not proportional to the amount of originalmaterial included; the explanation of the speaker’s latest theorem may fail toimprove his chance of creating a good impression.

An oft-quoted compromise between trying to be intelligible and trying toseem deep is this advice: address the first quarter of your talk to your high-schoolchemistry teacher, the second to a graduate student, the third to an educatedmathematician whose interests are different from yours, and the last to the spe-cialists. I have done my duty by reporting the formula, but I’d fail in my duty if Ididn’t warn that there are many who do not agree with it. A good public lectureshould be a work of art. It should be an architectural unit whose parts reinforce



each other in conveying the maximum possible amount of information—not a cam-paign speech that offers something to everybody, and more likely than not, endsby pleasing nobody. Make it simple, and you won’t go wrong. . .

Excerpt from: I Want to Be a Mathematician, p. 401, Springer-Verlag, NewYork (1985).

. . . As for working hard, I got my first hint of what that means whenCarmichael told me how long it took him to prepare a fifty-minute invited ad-dress. Fifty hours, he said: an hour of work for each minute of the final presen-tation. When many years later, six of us wrote our “history” paper (“Americanmathematics from 1940. . . ”), I calculated that my share of the work took about150 hours; I shudder to think how many manhours the whole group put in. Afew of my hours went toward preparing the lecture (as opposed to the paper). Italked it, the whole thing, out loud, and then, I talked it again, the whole thing,into a dictaphone. Then I listened to it, from beginning to end, six times—threetimes for spots that needed polishing (and which I polished before the next time),and three more times to get the timing right (and, in particular, to get the feelfor the timing of each part.) Once all that was behind me, and I had preparedthe transparencies, I talked the whole thing through one final rehearsal time (bymyself—no audience). That’s work. . .

On exposition

Excerpt from: Response from Paul Halmos on winning the Steele Prize for Expo-sition (1983).

Not long ago I ran across a reference to a publication titled A Method ofTaking Votes on More Than Two Issues. Do you know, or could you guess, who theauthor is? What about an article titled “On automorphisms of compact groups”?Who wrote that one? The answer to the first question is C.L. Dodgson, betterknown as Lewis Carroll, and the answer to the second question is Paul Halmos.

Lewis Carroll and I have in common that we both called ourselves mathe-maticians, that we both strove to do research, and that we both took very seriouslyour attempts to enlarge the known body of mathematical truths. To earn his liv-ing, Lewis Carroll was a teacher, and, just for fun, because he loved to tell stories,he wrote Alice’s Adventures in Wonerland. To earn my living, I’ve been a teacherfor almost fifty years, and, just for fun, because I love to organize and clarify,I wrote Finite Dimensional Vector Spaces. And what’s the outcome? I doubt ifas many as a dozen readers of these words have ever looked at either A Methodof Taking Votes. . . or “On automorphisms. . . ” but Lewis Carroll is immortal forthe Alice stories, and I got the Steele Prize for exposition. I don’t know what theReverend Mr. C.L. Dodgson thought about his fame, but, as for me, I was broughtup with the Puritan ethic: if something is fun, then you shouldn’t get recognized


16 J. Ewing

and rewarded for doing it. As a result, while, to be sure, I am proud and happy,at the same time I can’t help feeling just a little worried and guilty.

I enjoy studying, learning, coming to understand, and then explaining, but itdoesn’t follow that communicating what I know is always easy; it can be devilishlyhard. To explain something you must know not only what to put in, but also whatto leave out; you must know when to tell the whole truth and when to get theright idea across by telling a little white fib. The difficulty in exposition is notthe style, the choice of words—it is the structure, the organization. The wordsare important, yes, but the arrangement of the material, the indication of theconnections of its parts with each other and with other parts of mathematics, theproper emphasis that shows what’s easy and what deserves to be treated withcaution—these things are much more important. . .

On publishing

Excerpts from: “Four panel talks on publishing”, American MathematicalMonthly 82 (1975), 14–17.

. . . Let me remind you that most laws (with the exception only of the regula-tory statutes that govern traffic and taxes) are negative. Consider, as an example,the Ten Commandments. When Moses came back from Mount Sinai, he told uswhat to be by telling us, eight out of ten times, what not to do. It may thereforebe considered appropriate to say what not to publish. I warn you in advance thatall the principles that I was able to distill from interviews and from introspection,and that I’ll now tell you about, are a little false. Counterexamples can be foundto each one—but as directional guides the principles still serve a useful purpose.

First, then, do not publish fruitless speculations: do not publish polemicsand diatribes against a friend’s error. Do not publish the detailed working out ofa known principle. (Gauss discovered exactly which regular polygons are ruler-and-compass constructible, and he proved, in particular, that the one with 65537sides—a Fermat prime—is constructible; please do not publish the details of theprocedure. It’s been tried.)

Do not publish in 1975 the case of dimension 2 of an interesting conjecturein algebraic geometry, one that you don’t know how to settle in general, and thenfollow it by dimension 3 in 1976, dimension 4 in 1977, and so on, with dimensionk − 3 in 197k. Do not, more generally, publish your failures: I tried to prove so-and-so; I couldn’t; here it is—see?!

Adrian Albert used to say that a theory is worth studying if it has at leastthree distinct good hard examples. Do not therefore define and study a new class offunctions, the ones that possess left upper bimeasurably approximate derivatives,unless you can, at the very least, fulfill the good graduate student’s immediaterequest: show me some that do and show me some that don’t.

A striking criterion for how to decide not to publish something was offered bymy colleague John Conway. Suppose that you have just finished typing a paper.



Suppose now that I come to you, horns, cloven hooves, forked tail and all, andask: if I gave you $1,000,000, would you tear the paper up and forget it? If youhesitate, your paper is lost—do not publish it. That’s part of a more general rule:when in doubt, let the answer be no. . .

On research

Excerpt from: I Want to Be a Mathematician, pp. 321–322, Springer-Verlag,New York (1985).

Can anyone tell anyone else how to do research, how to be creative, how todiscover something new? Almost certainly not. I have been trying for a long timeto learn mathematics, to understand it, to find the truth, to prove a theorem, tosolve a problem—and now I am going to try to describe just how I went about it.The important part of the process is mental, and that is indescribable—but I canat least take a stab at the physical part.

Mathematics is not a deductive science—that’s a cliché. When you try toprove a theorem, you don’t just list the hypotheses, and then start to reason.What you do is trial and error, experimentation, guesswork. You want to find outwhat the facts are, and what you do is in that respect similar to what a laboratorytechnician does, but it is different in the degree of precision and information.Possibly philosophers would look on us mathematicians the same way we look onthe technicians, if they dared.

I love to do research, I want to do research, I have to do research, and I hateto sit down and begin to do research—I always try to put it off just as long as Ican.

It is important to me to have something big and external, not inside myself,that I can devote my life to. Gauss and Goya and Shakespeare and Paganini areexcellent, their excellence gives me pleasure, and I admire and envy them. Theywere also dedicated human beings. Excellence is for the few but dedication issomething everybody can have—and should have—and without it life is not worthliving.

Despite my great emotional involvement in work, I just hate to start doingit; it’s a battle and a wrench every time. Isn’t there something I can (must?) dofirst? Shouldn’t I sharpen my pencils, perhaps? In fact I never use pencils, butpencil sharpening has become the code phrase for anything that helps to postponethe pain of concentrated creative attention. It stands for reference searching inthe library, systematizing old notes, or even preparing tomorrow’s class lecture,with the excuse that once those things are out of the way I’ll really be able toconcentrate without interruption.

When Carmichael complained that as dean he didn’t have more than 20 hoursa week for research I marvelled, and I marvel still. During my productive yearsI probably averaged 20 hours of concentrated mathematical thinking a week, butmuch more than that was extremely rare. The rare exception came, two or three


18 J. Ewing

times in my life, when long ladders of thought were approaching their climax.Even though I never was dean of a graduate school, I seemed to have psychicenergy for only three or four hours of work, “real work”, each day; the rest of thetime I wrote, taught, reviewed, conferred, refereed, lectured, edited, travelled, andgenerally sharpened pencils all the ways I could think of. Everybody who doesresearch runs into fallow periods. During mine the other professional activities,down to and including teaching trigonometry, served as a sort of excuse for living.Yes, yes. I may not have proved any new theorems today, but at least I explainedthe law of sines pretty well, and I have earned my keep.

Why do mathematicians do research? There are several answers. The one Ilike best is that we are curious—we need to know. That is almost the same as“because we want to,” and I accept that—that’s a good answer too. There are,however, more answers, ones that are more practical.

On teaching

Excerpt from: “The problem of learning to teach”, American MathematicalMonthly 82 (1975), 466–476.

The best way to learn is to do; the worst way to teach is to talk.About the latter: did you ever notice that some of the best teachers of the

world are the worst lecturers? (I can prove that, but I’d rather not lose quite somany friends.) And, the other way around, did you ever notice that good lecturersare not necessarily good teachers? A good lecture is usually systematic, complete,precise—and dull; it is a bad teaching instrument. When given by such legendaryoutstanding speakers as Emil Artin and John von Neumann, even a lecture canbe a useful tool—their charisma and enthusiasm come through enough to inspirethe listener to go forth and do something—it looks like such fun. For most ordi-nary mortals, however, who are not so bad at lecturing as Wiener was—not sostimulating!—and not so good as Artin—and not so dramatic!—the lecture is aninstrument of last resort for good teaching.

My test for what makes a good teacher is very simple: it is the pragmatic oneof judging the performance by the product. If a teacher of graduate students con-sistently produces Ph.D.’s who are mathematicians and who create high-qualitynew mathematics, he is a good teacher. If a teacher of calculus consistently pro-duces seniors who turn into outstanding graduate students of mathematics, or intoleading engineers, biologists, or economists, he is a good teacher. If a teacher ofthird grade “new math” (or old) consistently produces outstanding calculus stu-dents, or grocery store check-out clerks, or carpenters, or automobile mechanics,he is a good teacher.

For a student of mathematics to hear someone talk about mathematics doeshardly any more good than for a student of swimming to hear someone talk aboutswimming. You can’t learn swimming techniques by having someone tell you where



to put your arms and legs; and you can’t learn to solve problems by having someonetell you to complete the square or to substitute sinu for y.

Can one learn mathematics by reading it? I am inclined to say no. Readinghas an edge over listening because reading is more active—but not much. Readingwith pencil and paper on the side is very much better—it is a big step in the rightdirection. The very best way to read a book, however, with, to be sure, pencil andpaper on the side, is to keep the pencil busy on the paper and throw the bookaway.

Having stated this extreme position, I’ll rescind it immediately. I know thatit is extreme, and I don’t really mean it—but I wanted to be very emphatic aboutnot going along with the view that learning means going to lectures and readingbooks. If we had longer lives, and bigger brains, and enough dedicated expertteachers to have a student/teacher ratio of 1/1, I’d stick with the extreme views—but we don’t. Books and lectures don’t do a good job of transplanting the factsand techniques of the past into the bloodstream of the scientist of the future—butwe must put up with a second best job in order to save time and money. But, andthis is the text of my sermon today, if we rely on lectures and books only, we aredoing our students and their students, a grave disservice. . .

Excerpt from: “The heart of mathematics”, American Mathematical Monthly87 (1980), 519–524.

. . . How can we, the teachers of today, use the problem literature? Our as-signed task is to pass on the torch of mathematical knowledge to the technicians,engineers, scientists, humanists, teachers, and, not least, research mathematiciansof tomorrow: do problems help?

Yes, they do. The major part of every meaningful life is the solution of prob-lems; a considerable part of the professional life of technicians, engineers, scientists,etc., is the solution of mathematical problems. It is the duty of all teachers, and ofteachers of mathematics in particular, to expose their students to problems muchmore than to facts. It is, perhaps, more satisfying to stride into a classroom and givea polished lecture on the WeierstrassM -test than to conduct a fumble-and-blundersession that ends in the question: “Is the boundedness assumption of the test neces-sary for its conclusion?” I maintain, however, that such a fumble session, intendedto motivate the student to search for a counterexample, is infinitely more valuable.

I have taught courses whose entire content was problems solved by students(and then presented to the class). The number of theorems that the studentsin such a course were exposed to was approximately half the number that theycould have been exposed to in a series of lectures. In a problem course, however,exposure means the acquiring of an intelligent questioning attitude and of sometechnique for plugging the leaks that proofs are likely to spring; in a lecture course,exposure sometimes means not much more than learning the name of a theorem,being intimidated by its complicated proof, and worrying about whether it wouldappear on the examination.


20 J. Ewing

. . . Many teachers are concerned about the amount of material they mustcover in a course. One cynic suggested a formula; since, he said, students on theaverage remember only about 40% of what you tell them, the thing to do is tocram into each course 250% of what you hope will stick. Glib as that is, it probablywould not work.

Problem courses do work. Students who have taken my problem courses wereoften complimented by their subsequent teachers. The compliments were on theiralert attitude, on their ability to get to the heart of the matter quickly, and ontheir intelligently searching questions that showed that they understood what washappening in class. All this happened on more than one level, in calculus, in linearalgebra, in set theory, and, of course, in graduate courses on measure theory andfunctional analysis.

Why must we cover everything that we hope students will ultimately learn?Even if (to stay with an example already mentioned) we think that the WeierstrassM -test is supremely important, and that every mathematics student must knowthat it exists and must understand how to apply it—even then a course on thepertinent branch of analysis might be better for omitting it. Suppose that thereare 40 such important topics that a student must be exposed to in a term. Doesit follow that we must give 40 complete lectures and hope that they will all sinkin? Might it not be better to give 20 of the topics just a ten-minute mention (thename, the statement, and an indication of one of the directions in which it can beapplied), and to treat the other 20 in depth, by student-solved problems, student-constructed counterexamples, and student-discovered applications? I firmly believethat the latter method teaches more and teaches better. Some of the materialdoesn’t get covered but a lot of it gets discovered (a telling old pun that deservesto be kept alive), and the method thereby opens doors whose very existence mightnever have been suspected behind a solidly built structure of settled facts. As forthe Weierstrass M -test, or whatever was given short shrift in class—well, booksand journals do exist, and students have been known to read them in a pinch. . .

On mathematics

Excerpt from: “Mathematics as a creative art”, American Scientist 56 (1968),375–389.

Do you know any mathematicians—and, if you do, do you know anythingabout what they do with their time? Most people don’t. When I get into a conver-sation with the man next to me in a plane, and he tells me that he is somethingrespectable like a doctor, lawyer, merchant or dean, I am tempted to say that I amin roofing and siding. If I tell him that I am a mathematician, his most likely replywill be that he himself could never balance his check book, and it must be fun to bea whiz at math. If my neighbor is an astronomer, a biologist, a chemist, or any otherkind of natural or social scientist, I am, if anything, worse off—this man thinks heknows what a mathematician is, and he is probably wrong. He thinks that I spend



my time (or should) converting different orders of magnitude, comparing binomialcoefficients and powers of 2, or solving equations involving rates of reactions.

C.P. Snow points to and deplores the existence of two cultures; he worriesabout the physicist whose idea of modern literature is Dickens, and he chides thepoet who cannot state the second law of thermodynamics. Mathematicians, inconverse with well-meaning, intelligent, and educated laymen (do you mind if Irefer to all nonmathematicians as laymen?) are much worse off than physicists inconverse with poets. It saddens me that educated people don’t even know that mysubject exists. There is something that they call mathematics, but they neitherknow how the professionals use the word, nor can they conceive why anybodyshould do it. It is, to be sure, possible that an intelligent and otherwise educatedperson doesn’t know that egyptology exists, or haematology, but all you have totell him is that it does, and he will immediately understand in a rough generalway why it should and he will have some empathy with the scholar of the subjectwho finds it interesting.

Usually when a mathematician lectures, he is a missionary. Whether he istalking over a cup of coffee with a collaborator, lecturing to a graduate classof specialists, teaching a reluctant group of freshman engineers, or addressing ageneral audience of laymen—he is still preaching and seeking to make converts. Hewill state theorems and he will discuss proofs and he will hope that when he is donehis audience will know more mathematics than they did before. My aim today isdifferent—I am not here to proselytize but to enlighten—I seek not converts butfriends. I do not want to teach you what mathematics is, but only that it is.

I call my subject mathematics—that’s what all my colleagues call it, all overthe world—and there, quite possibly, is the beginning of confusion. The wordcovers two disciplines—many more, in reality, but two, at least two, in the samesense in which Snow speaks of two cultures. In order to have some words withwhich to refer to the ideas I want to discuss, I offer two temporary and ad hocneologisms. Mathematics, as the work is customarily used, consists of at least twodistinct subjects, and I propose to call them mathology and mathophysics. Roughlyspeaking, mathology is what is called pure mathematics, and mathophysics iscalled applied mathematics, but the qualifiers are not emotionally strong enoughto disguise that they qualify the same noun. If the concatenation of syllables I chosehere reminds you of other words, no great harm will be done; the rhymes alluded toare not completely accidental. I originally planned to entitle this lecture somethinglike “Mathematics is an art,” or “Mathematics is not a science,” and “Mathematicsis useless,” but the more I thought about it the more I realized that I mean that“Mathology is an art,” “Mathology is not a science,” and “Mathology is useless.”When I am through, I hope you will recognize that most of you have known aboutmathophysics before, only you were probably calling it mathematics; I hope thatall of you will recognize the distinction between mathology and mathophysics; andI hope that some of you will be ready to embrace, or at least applaud, or at thevery least, recognize mathology as a respectable human endeavor.


22 J. Ewing

In the course of the lecture I’ll have to use many analogies (literature, chess,painting), each imperfect by itself, but I hope that in their totality they will serveto delineate what I want delineated. Sometimes in the interest of economy of time,and sometimes doubtless unintentionally, I’ll exaggerate; when I’m done, I’ll beglad to rescind anything that was inaccurate or that gave offense in any way. . .

Mathematics is abstract thought, mathematics is pure logic, mathematics iscreative art. All these statements are wrong, but they are all a little right, andthey are all nearer the mark than “mathematics is numbers” or “mathematicsis geometric shapes”. For the professional pure mathematician, mathematics isthe logical dovetailing of a carefully selected sparse set of assumptions with theirsurprising conclusions via a conceptually elegant proof. Simplicity, intricacy, andabove all, logical analysis are the hallmark of mathematics.

The mathematician is interested in extreme cases—in this respect he is likethe industrial experimenter who breaks lightbulbs, tears shirts, and bounces carson ruts. How widely does a reasoning apply, he wants to know, and what happenswhen it doesn’t? What happens when you weaken one of the assumptions, or underwhat conditions can you strengthen one of the conclusions? It is the perpetualasking of such questions that makes for broader understanding, better technique,and greater elasticity for future problems.

Mathematics—this may surprise or shock you some—is never deductive inits creation. The mathematician at work makes vague guesses, visualizes broadgeneralizations, and jumps to unwarranted conclusions. He arranges and rearrangeshis ideas, and he becomes convinced of their truth long before he can write downa logical proof. The conviction is not likely to come early—it usually comes aftermany attempts, many failures, many discouragements, many false starts. It oftenhappens that months of work result in the proof that the method of attack theywere based on cannot possibly work and the process of guessing, visualizing, andconclusion-jumping begins again. A reformulation is needed and—and this toomay surprise you—more experimental work is needed. To be sure, by “experimentalwork” I do not mean test tubes and cyclotrons. I mean thought-experiments. Whena mathematician wants to prove a theorem about an infinite-dimensional Hilbertspace, he examines its finite-dimensional analogue, he looks in detail at the 2- and3-dimensional cases, he often tries out a particular numerical case, and he hopesthat he will gain thereby an insight that pure definition-juggling has not yielded.The deductive stage, writing the result down, and writing down its rigorous proofare relatively trivial once the real insight arrives; it is more like the draftsman’swork, not the architect’s. . .

The mathematical fraternity is a little like a self-perpetuating priesthood.The mathematicians of today train the mathematicians of tomorrow and, in effect,decide whom to admit to the priesthood. Most people do not find it easy to join—mathematical talent and genius are apparently exactly as rare as talent and geniusin paint and music—but anyone can join, everyone is welcome. The rules arenowhere explicitly formulated, but they are intuitively felt by everyone in theprofession. Mistakes are forgiven and so is obscure exposition—the indispensable



requisite is mathematical insight. Sloppy thinking, verbosity without content, andpolemic have no role, and—this is to me one of the most wonderful aspects ofmathematics—they are much easier to spot than in the nonmathematical fields ofhuman endeavor (much easier than, for instance, in literature among the arts, inart criticism among the humanities, and in your favorite abomination among thesocial sciences).

Although most of mathematical creation is done by one man at a desk, at ablackboard, or taking a walk, or, sometimes, by two men in conversation, mathe-matics is nevertheless a sociable science. The creator needs stimulation while he iscreating and he needs an audience after he has created. Mathematics is a sociablescience in the sense that I don’t think it can be done by one man on a desert island(except for a very short time), but it is not a mob science, it is not a team science. Atheorem is not a pyramid; inspiration has never been known to descend on a com-mittee. A great theorem can no more be obtained by a “project” approach than agreat painting: I don’t think a team of little Gausses could have obtained the the-orem about regular polygons under the leadership of a rear admiral anymore thana team of little Shakespeares could have written Hamlet under such conditions. . .

On pure and applied

Excerpt from: “Applied mathematics is bad mathematics”, pp. 9–20, appearingin Mathematics Tomorrow, edited by Lynn Steen, Springer-Verlag, New York(1981).

It isn’t really (applied mathematics, that is, isn’t really bad mathematics),but it’s different.

Does that sound as if I had set out to capture your attention, and, havingsucceeded, decided forthwith to back down and become conciliatory? Nothing ofthe sort! The “conciliatory” sentence is controversial, believe it or not; lots ofpeople argue, vehemently, that it (meaning applied mathematics) is not differentat all, it’s all the same as pure mathematics, and anybody who says otherwise isprobably a reactionary establishmentarian and certainly wrong.

If you’re not a professional mathematician, you may be astonished to learnthat (according to some people) there are different kinds of mathematics, and thatthere is anything in the subject for anyone to get excited about. There are; andthere is; and what follows is a fragment of what might be called the pertinentsociology of mathematics: what’s the difference between pure and applied, howdo mathematicians feel about the rift, and what’s likely to happen to it in thecenturies to come. . .

The pure and applied distinction is visible in the arts and in the humani-ties almost as clearly as in the sciences: witness Mozart versus military marches,Rubens versus medical illustrations, or Virgil’s Aeneid versus Cicero’s Philippics.Pure literature deals with abstractions such as love and war, and it tells aboutimaginary examples of them in emotionally stirring language. Pure mathematicsdeals with abstractions such as the multiplication of numbers and the congruence


24 J. Ewing

of triangles, and it reasons about Platonically idealized examples of them withintellectually convincing logic.

There is, to be sure, one sense of the word in which all literature is “applied”.Shakespeare’s sonnets have to do with the everyday world, and so does Tolstoy’sWar and Peace, and so do Caesar’s commentaries on the wars he fought; all startfrom what human beings see and hear, and all speak of how human beings moveand feel. In that same somewhat shallow sense all mathematics is applied. It allstarts from sizes and shapes (whose study leads ultimately to algebra and geom-etry), and it reasons about how sizes and shapes change and interact (and suchreasoning leads ultimately to the part of the subject that the professionals callanalysis).

There can be no doubt that the fountainhead, the inspiration, of all literatureis the physical and social universe we live in, and the same is true about mathe-matics. There is no doubt that the physical and social universe daily affects eachmusician, and painter, and writer, and mathematician, and that therefore a partat least of the raw material of the artist is the work of facts and motions, sightsand sounds. Continual contact between the work and art is bound to change thelatter, and perhaps even to improve it.

The ultimate goal of “applied literature”, and of applied mathematics, isaction. A campaign speech is made so as to cause you to pull the third lever on avoting machine rather than the fourth. An aerodynamic equation is solved so as tocause a plane wing to lift its load fast enough to avoid complaints from the homeowners near the airport. These examples are crude and obvious; there are subtlerones. If the biography of a candidate, a factually correct and honest biography,does not directly mention the forthcoming election, is it then pure literature?If a discussion of how mathematically idealized air flows around moving figuresof various shapes, a logically rigorous and correct discussion, does not mentionairplanes or airports, is it then pure mathematics? And what about the in-betweencases: the biography that, without telling lies, is heavily prejudiced; and the treatiseon aerodynamics that, without being demonstrably incorrect, uses cost-cuttingrough approximations—are they pure or applied?. . .

To confuse the issue still more, pure mathematics can be practically usefuland applied mathematics can be artistically elegant. Pure mathematicians, try-ing to understand involved logical and geometrical interrelations, discovered thetheory of convex sets and the algebraic and topological study of various classesof functions. Almost as if by luck, convexity has become the main tool in linearprogramming (an indispensable part of modern economic and industrial practice),and functional analysis has become the main tool in quantum theory and particlephysics. The physicist regards the applicability of von Neumann algebras (a part offunctional analysis) to elementary particles as the only justification of the former;the mathematician regards the connections as the only interesting aspect of thelatter. De gustibus non disputandum est?

Just as pure mathematics can be useful, applied mathematics can be morebeautifully useless than is sometimes recognized. Applied mathematics is not en-



gineering; the applied mathematician does not design airplanes or atomic bombs.Applied mathematics is an intellectual discipline, not a part of industrial technol-ogy. The ultimate goal of applied mathematics is action, to be sure, but, beforethat, applied mathematics is a part of theoretical science concerned with the gen-eral principles behind what makes planes fly and bombs explode. . .

The deepest assertion about the relation between pure and applied mathe-matics that needs examination is that it is symbiotic, in the sense that neither cansurvive without the other. Not only, as is universally admitted, does the appliedneed the pure, but, in order to keep from becoming inbred, sterile, meaningless,and dead, the pure needs the revitalization and the contact with reality that onlythe applied can provide. . .

On being a mathematician

Excerpt from: I Want to Be a Mathematician, p. 400, Springer-Verlag, NewYork (1985).

It takes a long time to learn to live—by the time you learn your time is gone.I spent most of a lifetime trying to be a mathematician—and what did I learn?What does it take to be one? I think I know the answer: you have to be born right,you must continually strive to become perfect, you must love mathematics morethan anything else.

Born right? Yes. To be a scholar of mathematics you must be born with tal-ent, insight, concentration, taste, luck, drive, and the ability to visualize and guess.For teaching you must in addition understand what kinds of obstacles learners arelikely to place before themselves, and you must have sympathy for your audience,dedicated selflessness, verbal ability, clear style, and expository skill. To be able,finally, to pull your weight in the profession with the essential clerical and admin-istrative jobs, you must be responsible, conscientious, careful, and organized—ithelps if you also have some qualities of leadership and charisma.

You can’t be perfect, but if you don’t try, you won’t be good enough.To be a mathematician you must love mathematics more than family, reli-

gion, money, comfort, pleasure, glory. I do not mean that you must love it to theexclusion of family, religion, and the rest, and I do not mean that if you do love it,you’ll never have any doubts, you’ll never be discouraged, you’ll never be ready tochuck it all and take up gardening instead. Doubts and discouragements are part oflife. Great mathematicians have doubts and get discouraged, but usually they can’tstop doing mathematics anyway, and, when they do, they miss it very deeply. . .

John EwingMath for America160 Fifth Ave, 8th flNew York, NY 10010, USAe-mail: [email protected]



Obituary: Paul Halmos, 1916–2006

Heydar Radjavi and Peter Rosenthal

Paul Halmos, one of the most influential mathematicians of the last half of thetwentieth century, died at the age of ninety on October 2, 2006. Paul wrote “To bea mathematician you must love mathematics more than family, religion, money,comfort, pleasure, glory.” Paul did love mathematics. He loved thinking about it,talking about it, giving lectures and writing articles and books. Paul also lovedlanguage, almost as much as he loved mathematics. That is why his books andexpository articles are so wonderful. Paul took Hardy’s famous dictum that “thereis no permanent place in the world for ugly mathematics” very seriously: he refor-mulated and polished all the mathematics that he wrote and lectured about, andpresented it in the most beautiful way.

Irving Kaplansky, the great Canadian mathematician who also died in 2006(at the age of eighty-nine), wrote “Paul Halmos is the wittiest person I know.”Many quotations from Paul’s writing illustrating Kaplansky’s statement can befound on the internet (just Google “Paul Halmos quotations”). Here are some:

– You can’t be perfect, but if you don’t try, you won’t be good enough.– If you have to ask, you shouldn’t even ask.– Once the problem is solved, its repetitive application has as much to do with

mathematics as the work of a Western Union messenger boy has to do withMarconi’s genius.

– The criterion for quality is beauty, intricacy, neatness, elegance, satisfaction,appropriateness – all subjective, but somehow mysteriously shared by all.

– There is no Berlitz course for the language of mathematics; apparently theonly way to learn it is to live with it for years.

– The recommendations I have been making are based partly on what I do, moreon what I regret not having done, and most on what I wish others had donefor me.

– Almost everybody’s answer to “What to publish?” can be expressed in eitherone word – “less” – or two words – “good stuff”.

This obituary is reprinted with the kind permission of the Canadian Mathematical Society fromVolume 39 issue 2 (March, 2007) of the CMS Notes.


28 H. Radjavi and P. Rosenthal

Man-Duen Choi put together a number of titles of Halmos’s writings to forma cute narrative – see “A Postscript” on page 799 of volume 103 (1996) of theAmerican Mathematical Monthly.

Paul liked to be provocative. He wrote, for example, “The best way to learnis to do; the worst way to teach is to talk.” He did follow this with “Having statedthis extreme position, I’ll rescind it immediately. I know that it is extreme, andI don’t really mean it – but I wanted to be very emphatic about not going alongwith the view that learning means going to lectures and reading books.” However,his explanation did not mollify some people who were very proud of their abilityto lecture.

Perhaps Paul’s most provocative comment in print (those who had the plea-sure of participating in discussions with him heard even more provocative state-ments) was his title for an article published in 1981: “Applied Mathematics isBad Mathematics.” Although Paul began the article with “It isn’t really (appliedmathematics, that is, isn’t really bad mathematics), but it’s different,” the titleangered many applied mathematicians.

Paul made fundamental contributions to ergodic theory and measure theory(his classic books “Lectures on Ergodic Theory” and “Measure Theory”, and manypapers) and to algebraic logic (see “Algebraic Logic” and, with S. Givant, “Logicas Algebra”). His book “Naive Set Theory” is a beautiful exposition of axiomatic(Zermelo-Fraenkel) set theory, in spite of its “naive” title. But Paul’s most impor-tant contributions to research in mathematics, at least from our prejudiced pointof view, were to the theory of operators on Hilbert space.

Paul created and led a vigorous school of operator theory. He introducedcentral concepts such as unitary dilations, subnormal operators and quasitriangu-larizability, and proved the fundamental theorems about them. These, and otherconcepts he developed, became major subjects of research; there is now a largebody of knowledge about each of these topics.

Paul had extraordinary ability to discover the central questions concerninga large number of different aspects of operator theory. In particular, his famousarticle “Ten problems in Hilbert space” shaped a great deal of subsequent researchin operator theory and in C∗-algebras. As Berkeley mathematician Don Sarason(who competes with Errett Bishop for the title “Most-distinguished of Halmos’twenty-one Ph.D. students”) wrote, in his introduction to Paul’s selected works,“Halmos embodies the ideal mixture of researcher and teacher. In him, each roleis indistinguishable from the other. Perhaps that is the key to his remarkableinfluence.”

Paul wrote what he termed “an automathography”, a fascinating book enti-tled “I Want to be a Mathematician.” This is a mathematical autobiography, andcontains much advice that is very useful to all mathematicians and to all thosewho aspire to be mathematicians. Towards the end of that book, Paul evaluateshis career as follows: “I was, in I think decreasing order of quality, a writer, aneditor, a teacher, and a research mathematician.”


Obituary: Paul Halmos, 1916–2006 29

Paul’s self-evaluation may be quite accurate, but it is important to under-stand how high a standard he was setting for himself. He was certainly as good amathematical writer as ever existed. As an editor, Paul played a central role in de-veloping several of the series of mathematics books published by Springer-Verlag,as well as in editing several journals. Virtually everyone who ever heard him lec-ture will testify that his lectures were maximally interesting, clear and inspiring.(Luckily, several videotapes of Paul’s lectures can be purchased from the A.M.S.and the M.A.A.). Moreover, Paul’s total contribution to research in mathematicsis very impressive.

As Paul wrote, “it takes a long time to learn to live – by the time you learnyour time is gone.” However, one can learn much about living from others, andPaul taught many mathematicians a huge amount, about mathematics and aboutlife. For instance, Paul wrote “I like to start every course I teach with a problem.”Those who wish to follow Paul’s example in this respect can use Paul’s book“Problems for Mathematicians Young and Old”; it contains a surprising variety ofbeautiful problems from a variety of areas of mathematics.

Paul advised efficiency in all tasks: if a letter has to be answered, or a reviewhas to be written for Mathematical Reviews, do it right away, rather than thinkingfor months “I’d better get to that.” Good advice but, we have to confess, we failedto follow it in the writing of this obituary. It doesn’t take a psychoanalyst to figureout why we failed: we wanted to postpone this last goodbye to Paul Halmos. Butthe time has now come. Goodbye, Paul; thanks very much for so much.

Heydar RadjaviDepartment of MathematicsUniversity of WaterlooWaterlooOntario N2L 3G1, Canadae-mail: [email protected]

Peter RosenthalDepartment of MathematicsUniversity of TorontoTorontoOntario M5S 3G3, Canadae-mail: [email protected]



Mathematical Review of“How to Write Mathematics” ∗

by P.R. Halmos

G. Piranian

Halmos delivers a vigorous piece of his mind on the craft of writing. His principalmessage concerns the spiral plan, whose wide adoption would certainly raise thestandards of mathematical prose in our books and journals. Lest readers deceivethemselves and believe that they need not go to the original paper, I abstain fromdescribing the plan.

In addition to giving counsel on global strategy, Halmos points out manytactical devices by which authors can bring their manuscripts nearer to unobtrusiveperfection. He expresses his displeasure over some barbarisms, for example, thehanging theorem, the thoughtless misuse of “given” and “any”, and the sloppyconstruction “if . . . , then . . . if . . . ”. He argues against the display of cumbersomesymbolic messages that only a machine or a fool would decode, and he looksafter many fine points, such as the hyphen in the compound adjective “measure-preserving”. He even expresses his contempt for the pedantic copy-editor who inthe preceding sentence would insist on putting the second quotation mark afterthe period.

This review is not a catalogue of what the paper offers. It merely serves noticethat a mathematician with an eminently successful personal style has describedhis technique of writing, and that whoever pays heed will profit.

∗ Enseignement Math. (2) 16 1970 123–152c© American Mathematical Society 1972, 2009.



Publications of Paul R. Halmos

The publications are listed chronologically, articles and books separately. Not listedare translations of Paul’s articles and books, of which there have been many (inFrench, German, Bulgarian, Russian, Czech, Polish, Finnish, Catalan).

Research and Expository Articles

(1939-1) On a necessary condition for the strong law of large numbers. Ann. ofMath. (2) 40 (1939), 800–804.

(1941-1) Statistics, set functions, and spectra. Rec. Math. [Mat. Sbornik] N.S. 9(51) (1941), 241–248.

(1941-2) The decomposition of measures. Duke Math. J. 8 (1941), 386–392.

(1942-1) The decomposition of measures. II. Duke Math. J. 9 (1942), 43–47(with W. Ambrose and S. Kakutani).

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